Monday, February 29, 2016

Imagine a horse-race (i.e. Benford Law Stakes) with the following
distribution of win probabilities.
This toy example is not as unrealistic as you might expect at first
glance. Look at the very good approximation by Benford's Law of Starting
Price position for roughly 20,000 GB flat races 2004-2013 inclusive.
Then, in simplest terms, the inherent uncertainty of the Benford Law
Stakes race outcome is best represented by Shannon's
Entropy: H(x) = -SUM((x)*log(x)) = 2.87, which number also suggests
(under optimal conditions) the minimum number of yes/no questions (i.e. 3)
the handicapper should ask himself to identify a potential winner. Taking
our lead from Shannon-Fano
Coding, we should iteratively divide the entrants into two
approximately equal groups of win probabilities (i.e. 50%) and use Pairwise
Comparison to eliminate the non-contenders using at most four
questions. Once again, this restriction is not as unrealistic as it might first
appear. Slovic And Corrigan (1973), in a study of expert handicappers,
found that with only five items of information the handicappers'
confidence was well calibrated with their accuracy but that they became
overconfident as additional information was received. This finding was
confirmed in a follow-up study by Tsai
et al (2008).